Is there a name for an ordered algebraic structure that abstracts the non-negative extended integers (extended in the sense that they include $\infty$), i.e. is there a name for a partially-ordered, abelian monoid with the additional requirement that every $x$, with the possible exception of the maximum, has an inverse? To put it differently, is there a name for a partially-ordered abelian monoid, which, excluding the maximum element (if any), forms a group?
2026-03-25 15:50:21.1774453821
Is there a name for an ordered algebraic structure that models the extended integers (i.e. enhanced with infinity)?
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It seems you are looking for partially ordered groups with a top element. Also known as po-groups --c.f. po-sets.
These are usually defined as sets having an order and a group operation such that the operation is monotone; i.e., order-preserving.
There are also po-monoids and it seems your particular interest is the case of commutative po-monoids with a top element.